扭结Zhu理论的对偶理论

中文摘要 AI 辅助

本文研究了分次顶点算子余代数$V$和有限阶自同构$g\in \Aut V$的可容许$g$-扭结$V$-余模的概念。我们证明了$\mathcal {M}$是可容许$g$-扭结$V$-余模当且仅当其分次对偶$\mathcal {M}'$是顶点算子代数$V'$的可容许$g^{-1}$-扭结$V'$-模。然后我们建立了扭结Zhu理论的对偶理论，即对于分次顶点算子余代数$V$，存在一个余结合余代数$C_g(V)$，使得任何可容许$g$-扭结$V$-余模都给出一个$C_g(V)$-余模，反之亦然。我们还证明了$V$是$g$-余有理的（即每个可容许$g$-扭结$V$-余模都是完全可约的）当且仅当其顶点算子代数对偶$V'$是$g$-有理的。

英文摘要

The notion of admissible $g$-twisted $V$-comodules is investigated for a graded vertex operator coalgebra $V$ and a finite order automorphism $g\in \Aut V$. We prove that $\mathcal {M}$ is an admissible $g$-twisted $V$-comodule if and only if its graded dual $\mathcal {M}'$ is an admissible $g^{-1}$-twisted $V'$-module for vertex operator algebra $V'$. Then we establish a dual theory of twisted Zhu's theory, i.e., there is a coassociative coalgebra $C_g(V)$ for a graded vertex operator coalgebra $V$, any admissible $g$-twisted $V$-comodule gives a $C_g(V)$-comodule, and vice versa. We also prove that $V$ is $g$-corational, which means every admissible $g$-twisted $V$-comodule is completely reducible, if and only if its dual vertex operator algebra $V'$ is $g$-rational.

发表机构

• Northwest University（西北大学）

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